Question

If vector a=2i-j+k, b=i-3j+k,c= -2i+j-3k,d=3i+2j+5k find the p,q,r such that d=pa+qb+rc.
Please help me how to do it...!!!

Answer 1

I will use the notation
a = (2,-1,1) , b= (1,-3,1) , c=(-2,1,-3) and d=(3,2,5)

so
2p + q - 2r = 3
-p - 3q + r = 2
-2p + q - 3r = 5

add 1st and 3rd:
2q - 5r = 8 **

double the 2nd , then add to the first:
-5q = 7
q = -7/5

into **
-14/5 - 5r = 8
-5r = 54/5
r = -54/25

into the 1st:
2p - 7/5 +108/25 = 3
2p = 2/25
p = 1/25


Wolfram confirms:
http://www.wolframalpha.com/input/?i=solve+2p+%2B+q+-+2r+%3D+3,+-p+-+3q+%2B+r+%3D+2,+-2p+%2B+q+-+3r+%3D+5

Answer 2

oops, just noticed my third equation is wrong
Here we go again:

I will use the notation
a = (2,-1,1) , b= (1,-3,1) , c=(-2,1,-3) and d=(3,2,5)

so
2p + q - 2r = 3
-p - 3q + r = 2
p + q - 3r = 5

add the 2nd and 3rd:
-2q - 2r = 7
2q +2r = -7 **

double the 2nd , then add to the first:
-5q = 7
q = -7/5

into **
-14/5 + 2r = -7
2r = -21/5
r = -21/10

into the 1st:
2p -7/5 + 42/10 = 3
2p = 1/5
p = 1/10

check in original setup

(1/10)(2,-1,1) + (-7/5)(1,-3,1) + (-21/10)(-2,1,-3)
= (2/10,-1/10,-1/10) + (-7/5,21/5,-7/5) + (42/10,-21/10,63/10)
= (3, 2 , 5)

Answer 3

@Reiny...Thank you :)